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Stability of a functional equation for square root spirals. (English) Zbl 1016.39020

The authors study a concrete case of the stability (in the sense of Hyers-Ulam-Rassias) of the equation of the square root spiral: \[ f( \sqrt {r^2 +1}) = f (r) + \arctan \tfrac 1r.\tag{1} \] The main result is the following. If \( f: [1, \infty) \to [0, \infty) \) satisfies, for all \( r \geq 1\), \( |f (\sqrt {r^2 +1})- f(r)- \arctan {1 \over r} |\) \(\leq g(r)\), where \( g: [1 , \infty) \to [0, \infty)\) is such that \( G(r) : = \sum_{k=0}^{\infty} g (\sqrt {r^2 + k}) < \infty \), then there exists a unique solution \(F\) of (1) such that, for all \( r \geq 1\), \[ |F(r)- f(r) |\leq G(r). \]

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
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